Type your answer with the precision of 2 decimal places.

Let X be a chi-square random variable with 13 degrees of freedom. Find the standard deviation of X.

Question

Type your answer with the precision of 2 decimal places.

Let X be a chi-square random variable with 13 degrees of freedom. Find the standard deviation of X.

BlackTom AI · Accepted Answer

For a chi-square distribution with ν degrees of freedom, the variance equals 2ν, so the standard deviation is sqrt(2ν).

1. Relevant knowledge points and concepts
- Chi-square(ν) has mean μ = ν and variance Var(X) = 2ν.
- Standard deviation is the positive square root of the variance: σ = sqrt(Var(X)) = sqrt(2ν).

2. Complete reasoning process step by step
- Given ν = 13.
- Compute the variance using Var(X) = 2ν.
- Take the square root to obtain the standard deviation.

3. Calculations with intermediate steps
- Variance: Var(X) = 2 × 13 = 26.
- Standard deviation: σ = sqrt(26).
- Evaluate sqrt(26):
  - Note sqrt(25) = 5, so sqrt(26) = 5 × sqrt(26/25) = 5 × sqrt(1.04).
  - sqrt(1.04) ≈ 1.0198039, so σ ≈ 5 × 1.0198039 = 5.0990195...
- Round to two decimal places: σ ≈ 5.10.

4. Why this answer is correct
- The derivation follows directly from the known properties of the chi-square distribution: its variance is exactly 2ν; taking the square root yields sqrt(2ν). Plugging ν = 13 gives sqrt(26), numerically 5.099..., which correctly rounds to 5.10.

5. Why other plausible answers would be incorrect
- Using ν (13) as the standard deviation confuses mean with standard deviation.
- Using sqrt(ν) ≈ 3.61 is wrong because variance is not ν but 2ν.
- Using 26 would confuse variance with standard deviation (26 is the variance, not SD).
- Only sqrt(2ν) = sqrt(26) gives the correct standard deviation.

6. Conclusion and confirmation
Therefore the standard deviation of X (χ^2 with 13 degrees of freedom) is 5.10, which matches the provided answer.

Type your answer with the precision of 2 decimal places. Let X be a chi-square random variable with 13 degrees of freedom. Find the standard deviation of X.Short answer

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